Most of the existing papers for constructing coresets essentially had to bound the total importance of the related problem anyway. However, without Theorem REF , their proofs also had to deal with complicated terms that involved \(\varepsilon \)  and include sophisticated probability arguments. Essentially each paper had to re-prove in some sense the very involved proofs in [1]}, [2]} that were researched for decades, as well as the mathematics behind the usage of Definition REF . Beside the complicated proofs, the final bounds, the dependency on \(\varepsilon \)  and \(\delta \) , as well as the coreset construction algorithm, were usually sub-optimal compared to Theorem REF . On the contrary, bounding the total importance allows us to focus on a deterministic results (no \(\delta \)  involved) and the terms \(s(p)\)  can be approximated up to constant factors (and not \((1+\varepsilon )\) -factors).
This is demonstrated in Section .
